Estimation of the Feasible Solutions in Determining Parameters of Symmetrical Overlapping Peaks

Joseph Dubrovkin, Western Galilee College, Acre, Israel

First-order analytical signal obtained by analytical instruments (e.g., in spectrometry, chromatography, flow injection analysis and voltammetry) is usually represented as a combination of bell-shaped peaks, background and random noise. Since peak overlapping hinders accurate estimation of the peak parameters, analytical signals must be decomposed into pure peaks by fitting a priory-defined model to the measured data.

In the present study the squared norm of residuals in the curve-fitting was represented by quadratic form in the canonical form. Matrix of the quadratic form was estimated using the first-order Taylor polynomial expansion of the objective function. The Jacobian was obtained analytically and numerically. Unlike the traditional approach based on minimizing the norm of residuals, the relative errors of the peak parameters of the model (undetectable perturbations – UPs [1]) were estimated for a priory given norm of residuals. All calculations were performed in spherical coordinates. Correlations between UPs were taken into account. UPs set the boundaries of feasible solutions in determining parameters of overlapping peaks.

Numerical analysis of decomposition of simulated Gaussian single peak, doublets and triplets) allowed us to establish relationships between UPs and the model parameters.

As example, badly resolved visible absorption spectrum of the water solution of crystal violet was decomposed into three Gaussians. Two best-fit models were obtained. It was shown that if peak parameters are taken randomly from the confidence intervals estimated by the linear approximation, then a probability of "the best fitting" is only 62%. However, if the relative errors of the peak parameters are taken from the multivariate normal distribution which takes into account correlations between the errors then this probability is more than 99%. Physically based model of the crystal violet spectrum has been chosen.

[1] Dubrovkin, J.(2016) Chemom. Intell. Lab. Syst. 153, 9-21.